Alge-Tiles Making the Connection between the Concrete ↔ Symbolic

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Presentation on theme: "Alge-Tiles Making the Connection between the Concrete ↔ Symbolic"— Presentation transcript:

1 Alge-Tiles Making the Connection between the Concrete ↔ Symbolic(Alge-tiles) ↔ (Algebraic)

2 What are Alge-Tiles?Alge-Tiles are rectangular and square shapes (tiles) used to represent integers and polynomials.Examples: 1→1x →1x2 →

3 Objectives for this lessonUsing Alge-Tiles for the following:- Combining like terms- Multiplying polynomials- Factoring- Solving equationsAllow students to work in small groups when doing this lesson.

4 Construction of Alge-Tiles1 (let the side = one unit)For one unit tile:(it is a square tile)1Area = (1)(1) = 1x(unknown length therefore let it = x)For a 1x tile(it is a rectangular tile)1Side of unit tile = side of x tileArea = (1)(x) = 1xxSide of x2 tile = side of x tileFor x2 tile:(It is a square tile)Area = (x)(x) = x2xOther side of x2 tile = side of x tile

6 Part I: Combining Like TermsFor negative numbers use the other side of each tile (the white side)Use the Alge Tiles to represent the following:-2x →-4 →-3x - 4 →

7 Part I: Combining Like TermsRepresent “2x” with tilesRepresent “3” with tilesCan 2x tiles be combined with the tiles for 3 to make one of our three shapes? Why or why not?Therefore: simplify 2x + 3 =2x + 3 can’t be simplified any further (can’t touch this)

9 Part I: Combining Like TermsAfter mastering several questions where students were combing terms you could then pose the question to the class working in groups:“Is there a pattern or some kind of rule you can come up with that you can use in all situations when combining polynomials.”In conclusion, when combining like terms you can only combine terms that have the same tile shape (concrete) → Algebraic: Can combine like terms if they have the same variable and exponent.

10 Part II: Multiplying PolynomialsPrerequisites: Students were taught the distributive property and finding the area of a rectangle.Area(rectangle) = length x widthWhen multiplying polynomials the terms in each bracket represents the width or length of a rectangle.Find the area of a rectangle with sides 2 and 3. Two can be the width and 3 would be the length.The area of the rectangle would = (2)•(3) = 6

11 Part II: Multiplying PolynomialsWe will use tiles to find the answer. The same premise will be used as finding the area of a rectangle.Make the length = 3 tilesThe width = 2 tilesThe tiles form a rectangle, use other tiles to fill in the rectangleOnce the rectangle is filled in remove the sides and what is left is your answer in this case it is 6 or 6 unit tiles

15 Part II: Multiplying PolynomialsPattern: After mastering several questions where students were combing terms you could then pose the question to the class working in groups:“Is there a pattern or some kind of rule you can come up with that you can use in all situations when multiplying polynomials.”This can lead to a larger discussion where students can put forth their ideas.

16 Part III: FactoringOutcomes: Grade 9 – B9, B10, Grade 10 – B1, B3, C16Take an expression like 2x + 4 and use the rectangle to factor.You will go in reverse when being compared to multiplying polynomials. (make the rectangle to help find the sides)The factors will be the sides of the rectangleConstruct a rectangle using 2 ‘x’ tiles and 4 unit one tiles. This can be tricky until you get the hang of it.

17 Part III: FactoringNow make the sides; width and length of the rectangle using the alge-tiles.Side 1: (1x + 2)Side 2: (2)2x + 4 = (2)(1x +2)Remove the rectangle and what is left are the factors of 2x +4

19 Part III: Factoring x2 + 5x + 6 = (1x + 3) (1x + 2)Try factoring x2 + 5x + 6 (make rectangle)(1x + 3)**Hint: when the expression has x2, start with the x2 tile.Next, place the 6 unit tiles at the bottom right hand corner of the x2 tile. You will make a small rectangle with the unit tiles.(1x + 2)32Then add the x tiles where needed to complete the rectangleWhen the rectangle is finished examine it to see if the tiles combine to give you the original expression → x2 + 5x + 6x2 + 5x + 6 = (1x + 3) (1x + 2)Next make the sides for the rectangleRemove the rectangle and you have the factors. (1x + 3) (1x + 2)

20 Part III: Factoring What if someone tried the following:Factor: x2 + 5x + 6 (make rectangle)Start with the x2 tile, now make a rectangle with the 6 unit tiles.Now complete the rectangle using the x tiles.1When the rectangle is finished examine it to see if the tiles combine to give you the original expression → x2 + 5x + 66When the tiles are combined, the result is x2 + 7x + 6, where is the mistake?The unit tiles must be arranged in a rectangle so when the x tiles are used to complete the rectangle they will combine to equal the middle term, in this case 5x.

21 Factoring Have students try to factor more trinomials(refer to Alge-tile binder – Factoring section: F – 3b for additional questions)After mastering several questions where students were factoring trinomials you could then pose the question to the class :“Is there a pattern or some kind of rule you can come up with that you can use when factoring trinomials?”

22 Part III: Factoring (negatives)Try factoring: x2 - 1x – 6Start with x2 tile, then fill in the unit tiles in this case -6 which is 6 white unit tiles.Remember to make a rectangle at the bottom corner of the x2 tiles where the sides have to add to equal the coefficient of the middle term, -1.1x - 31x + 2-3Next fill in the x tiles to make the rectangle.2Now the rectangle is complete check to see if the tiles combine to equalx2 - 1x – 6.Therefore x2 - 1x – 6 = (x – 3) (x + 2)Fill in the sides and remove the rectangle to give you the factors.

26 Alge-Tile ConclusionAssessment: While students are working on question sheet handout, go around to each group and ask students to do some questions for you to demonstrate what they have learned.For practice refer to handout of questions for all four sections:Part I: Combining Like TermsPart II: Multiplying PolynomialsPart III: FactoringPart IV: Solving for an unknown(P.S. the answers are at the end)